Lemma 1 ([1]}) 
Given \(\lambda _1,...,\lambda _n\)  in \(\mathbf {R}\)  with \(\sum _{i=1}^n\lambda _i=0\)  and \(x_1,...,x_n\)  in \(\mathbf {R^d}\) . Then for \(\lambda _1x_1+\lambda _2x_2+...+\lambda _nx_n,\)  the terms with positive coefficients can be matched with the terms with negative coefficients in the sense that
\(\sum _{i=1}^n\lambda _ix_i=\sum _{i,c_i\ge 0,j,k\in [n]} c_{i}(x_{j}-x_{k})\text{ and }\sum _i c_i=\sum _{j,\lambda _j\ge 0}\lambda _j.\) 

